Fitting GP to lightcurve 80_ra271.352_dec-29.642_MAXIJ1803 in Stan

Notebook outlining the fitting of GP to thunderKAT lightcurve 80_ra271.352_dec-29.642_MAXIJ1803.

1 Light Curve

  • The light curve has \(N =\) 28 observations over a range of 198.79 days.
  • Observations are evenly spread over the time range.
  • The shortest gap between observations is 2.22 days.
  • The longest gap between observations is 14.67 days.
  • The mean flux density is \(\bar{y} =\) 0.0587 Jy.
  • The mean standard error in the observations is 0.0000624 Jy.
  • The observational noise is very small relative to the brightness of the observations.

2 SE Basic Model

  • Zero constant mean function.
  • Squared Exponential kernel function.
  • Homoskedastic noise.
  • Wide prior on observational noise, uninformed by observational noise estimates.

\[y \sim \mathcal{N}(f(x), \sigma_\textrm{noise}^2)\]

\[f \sim \mathcal{GP}(\boldsymbol{0}, k(x, x'))\]

\[k(x,x') = \eta^2 \exp\left\{ -\frac{1}{2}\frac{(x - x')^2}{\ell^2}\right\}\]

\[\ell \sim \mathrm{InvGamma}(5,5)\]

\[\eta \sim \mathcal{N}^+(0,1)\]

\[\sigma_\textrm{noise} \sim \mathcal{N}^+(0,1)\]

2.1 MCMC Results

 variable      mean    median       sd      mad       q5       q95    rhat
    eta     0.06162   0.04861  0.04928  0.02055  0.02711   0.13266 1.00067
    ell   164.05323 142.24100 85.46926 52.09634 83.40403 320.12535 1.00086
    sigma   0.00155   0.00153  0.00023  0.00024  0.00121   0.00196 1.00239
 ess_bulk ess_tail
     2465     1562
     2322     1841
     2873     2447

2.2 MCMC Plots

2.3 Posterior Predictive Samples

The fitted model has a very long lengthscale, comparable to the length of the observational window. The estimated observational noise has a standard deviation is two orders of magnitude greater than that recorded in the original data. The combination of these parameters has lead to a very smooth fit that passes through the middle of the observed data points rather than through any datapoints themselves.

2.4 PSD

3 SE Observational Errors Model

  • Zero constant mean function.
  • Squared exponential kernel function.
  • Heteroskedastic (Gaussian) noise.
  • Incorporate data on error in observations of each \(y_i\).

\[y_i \sim \mathcal{N}(f(x_i), \sigma_i^2)\]

\[f \sim \mathcal{GP}(\boldsymbol{0}, k(x, x'))\]

\[k(x,x') = \eta^2 \exp\left\{ -\frac{1}{2}\frac{(x - x')^2}{\ell^2}\right\}\]

\[\ell \sim \mathrm{InvGamma}(5,5)\]

\[\eta \sim \mathcal{N}^+(0,1)\]

\[\sigma_i \sim \mathcal{N}^+(\textrm{stderr}(y_i), \mathrm{Var}(\textrm{stderr}(\boldsymbol{y})))\]

3.1 MCMC Results

 variable      mean    median       sd      mad       q5       q95     rhat
 eta       0.045350  0.044251 0.007867 0.007204 0.034750  0.059706 0.999626
 ell      10.168749 10.187500 0.436035 0.420339 9.411075 10.853310 0.999804
 sigma[1]  0.000086  0.000086 0.000009 0.000009 0.000072  0.000100 1.000408
 ess_bulk ess_tail
     4127     3038
     4093     2572
     4849     2893

3.2 MCMC Plots

3.3 Posterior Predictive Samples

By including the observed observational errors for setting priors on the Gaussian noise of each observation, the fitted median passes through each of the observed points.

3.4 PSD

4 SE Non-zero flat mean function Model

  • Constant mean function, learned from observations.
  • Weak prior on mean function intercept.
  • Squared exponential kernel function.
  • Heteroskedastic (Gaussian) noise.
  • Incorporate data on error in observations of each \(y_i\).

\[y_i \sim \mathcal{N}(f(x_i), \sigma_i^2)\]

\[f \sim \mathcal{GP}(C, k(x, x'))\]

\[C \sim \mathcal{U}[-1,1]\]

\[k(x,x') = \eta^2 \exp\left\{ -\frac{1}{2}\frac{(x - x')^2}{\ell^2}\right\}\]

\[\ell \sim \mathrm{InvGamma}(5,5)\]

\[\eta \sim \mathcal{N}^+(0,1)\]

\[\sigma_i \sim \mathcal{N}^+(\textrm{stderr}(y_i), \mathrm{Var}(\textrm{stderr}(\boldsymbol{y})))\]

4.1 MCMC Results

 variable     mean   median       sd      mad       q5      q95     rhat
 eta      0.001595 0.001569 0.000231 0.000226 0.001264 0.002006 0.999994
 ell      0.978960 0.919284 0.358948 0.322083 0.512659 1.647210 1.000892
 C        0.058665 0.058667 0.000313 0.000300 0.058149 0.059188 0.999833
 sigma[1] 0.000086 0.000086 0.000009 0.000008 0.000071 0.000100 1.001164
 ess_bulk ess_tail
     5282     2601
     5617     2842
     3952     2478
     5717     2777

4.2 MCMC Plots

4.3 Posterior Predictive Samples

4.4 PSD

5 SE Fixed constant mean function Model

  • Constant mean function set at fixed value, e.g., mean of observations.
  • Squared exponential kernel function.
  • Heteroskedastic (Gaussian) noise.
  • Incorporate data on error in observations of each \(y_i\).

\[y_i \sim \mathcal{N}(f(x_i), \sigma_i^2)\]

\[f \sim \mathcal{GP}(C, k(x, x')),\quad C \in \mathbb{R}\]

\[k(x,x') = \eta^2 \exp\left\{ -\frac{1}{2}\frac{(x - x')^2}{\ell^2}\right\}\]

\[\ell \sim \mathrm{InvGamma}(5,5)\]

\[\eta \sim \mathcal{N}^+(0,1)\]

\[\sigma_i \sim \mathcal{N}^+(\textrm{stderr}(y_i), \mathrm{Var}(\textrm{stderr}(\boldsymbol{y})))\]

5.1 Mean Function = 0.061

 variable    mean  median      sd     mad      q5     q95    rhat ess_bulk
 eta      0.00291 0.00286 0.00040 0.00038 0.00235 0.00362 1.00174     5189
 ell      1.29514 1.08483 0.75875 0.47578 0.55316 2.97471 1.00279     4867
 sigma[1] 0.00009 0.00009 0.00001 0.00001 0.00007 0.00010 1.00105     5855
 ess_tail
     2379
     2620
     2817

5.2 Mean Function = 0.059

 variable     mean   median       sd      mad       q5      q95     rhat
 eta      0.001608 0.001581 0.000231 0.000221 0.001278 0.002020 1.000380
 ell      0.971589 0.911971 0.358267 0.319847 0.502400 1.644370 0.999937
 sigma[1] 0.000086 0.000085 0.000009 0.000009 0.000071 0.000100 1.001193
 ess_bulk ess_tail
     6311     2673
     5394     2855
     6013     2649

5.3 Mean Function = 0.056

 variable    mean  median      sd     mad      q5     q95    rhat ess_bulk
 eta      0.00319 0.00314 0.00044 0.00043 0.00255 0.00397 1.00035     6035
 ell      1.38000 1.13118 0.81861 0.54967 0.55854 3.25948 1.00267     4442
 sigma[1] 0.00009 0.00009 0.00001 0.00001 0.00007 0.00010 1.00250     5813
 ess_tail
     3143
     3252
     2901

6 Matern 3/2 kernel

  • Matern 3/2 covariance kernel
  • Zero constant mean function

\[y_i \sim \mathcal{N}(f(x_i), \sigma_i^2)\]

\[f \sim \mathcal{GP}(\boldsymbol{0}, k(x, x'))\]

\[k(x,x') = \eta^2 \left( 1 + \frac{\sqrt{3(x - x')^2}}{\ell}\right) \exp\left\{ -\frac{\sqrt{3(x - x')^2}}{\ell}\right\}\]

\[\ell \sim \mathrm{InvGamma}(5,5)\]

\[\eta \sim \mathcal{N}^+(0,1)\]

\[\sigma_i \sim \mathcal{N}^+(\textrm{stderr}(y_i), \mathrm{Var}(\textrm{stderr}(\boldsymbol{y})))\]

6.1 MCMC Results

 variable      mean    median        sd       mad        q5       q95     rhat
 eta       0.048637  0.044109  0.021800  0.012371  0.029688  0.080836 1.000238
 ell      60.439406 57.039200 17.662723 12.974381 40.831115 90.178535 1.000777
 sigma[1]  0.000086  0.000086  0.000009  0.000009  0.000071  0.000100 1.000203
 ess_bulk ess_tail
     4058     2363
     4186     2374
     8834     2546

6.2 MCMC Plots

6.3 Posterior Predictive Samples

6.4 PSD

7 SE + Matern 3/2 additive kernel

  • Sum of squared exponential and Matern 3/2 kernels
  • single output scale (marginal variance) hyperparameter
  • zero constant mean function

\[y_i \sim \mathcal{N}(f(x_i), \sigma_i^2)\]

\[f \sim \mathcal{GP}(\boldsymbol{0}, k(x, x'))\]

\[k(x,x') = \eta^2 \left[ \exp\left\{ -\frac{1}{2}\frac{(x - x')^2}{\ell_\mathrm{SE}^2}\right\} + \left( 1 + \frac{\sqrt{3(x - x')^2}}{\ell_\mathrm{M}}\right) \exp\left\{ -\frac{\sqrt{3(x - x')^2}}{\ell_\mathrm{M}}\right\} \right]\]

\[\ell_\mathrm{SE} \sim \mathrm{InvGamma}(5,5)\]

\[\ell_\mathrm{M} \sim \mathrm{InvGamma}(5,5)\]

\[\eta \sim \mathcal{N}^+(0,1)\]

\[\sigma_i \sim \mathcal{N}^+(\textrm{stderr}(y_i), \mathrm{Var}(\textrm{stderr}(\boldsymbol{y})))\]

7.1 MCMC Results

 variable     mean   median       sd      mad       q5      q95    rhat
 eta       0.00109  0.00086  0.00084  0.00045  0.00038  0.00246 1.00042
 ell_SE   44.26588 37.72915 26.17904 14.15305 21.78390 87.93888 1.00209
 ell_M    43.05689 41.35540 10.95611  9.51570 28.65622 62.88434 1.00089
 sigma[1]  0.00009  0.00009  0.00001  0.00001  0.00007  0.00010 1.00117
 ess_bulk ess_tail
     4254     2429
     4581     2554
     4192     2537
     8729     2493

7.2 MCMC Plots

7.3 Posterior Predictive Samples

7.4 PSD

8 SE + Matern 3/2 (2 output scales) additive kernel

  • Sum of squared exponential and Matern 3/2 kernels
  • One output scale (marginal variance) hyperparameter for each kernel
  • zero constant mean function

\[y_i \sim \mathcal{N}(f(x_i), \sigma_i^2)\]

\[f \sim \mathcal{GP}(\boldsymbol{0}, k(x, x'))\]

\[k(x,x') = \eta_\textrm{SE}^2 \exp\left\{ -\frac{1}{2}\frac{(x - x')^2}{\ell_\mathrm{SE}^2}\right\} + \eta^2_\textrm{M}\left( 1 + \frac{\sqrt{3(x - x')^2}}{\ell_\mathrm{M}}\right) \exp\left\{ -\frac{\sqrt{3(x - x')^2}}{\ell_\mathrm{M}}\right\}\]

\[\ell_\mathrm{SE} \sim \mathrm{InvGamma}(5,5)\]

\[\ell_\mathrm{M} \sim \mathrm{InvGamma}(5,5)\]

\[\eta_\textrm{SE} \sim \mathcal{N}^+(0,1)\]

\[\eta_\textrm{M} \sim \mathcal{N}^+(0,1)\]

\[\sigma_i \sim \mathcal{N}^+(\textrm{stderr}(y_i), \mathrm{Var}(\textrm{stderr}(\boldsymbol{y})))\]

8.1 MCMC Results

 variable      mean    median        sd       mad      q5        q95    rhat
 eta_SE     0.00502   0.00000   0.03622   0.00000 0.00000    0.01485 1.52781
 eta_M      0.00349   0.00120   0.02520   0.00177 0.00000    0.00964 1.52842
 ell_SE    50.34066   1.06769 139.32085   0.53340 0.54951  242.32025 1.52828
 ell_M    430.14665 365.39350 482.34979 334.13727 0.69897 1137.66900 1.52856
 sigma[1]   0.00009   0.00009   0.00001   0.00001 0.00007    0.00010 0.99977
 ess_bulk ess_tail
        7       31
        7       27
        7       28
        7       26
    10254     5487

8.2 MCMC Plots

8.3 Posterior Predictive Samples

8.4 PSD

9 SE x Matern 3/2 multiplicative kernel

\[y_i \sim \mathcal{N}(f(x_i), \sigma_i^2)\]

\[f \sim \mathcal{GP}(\boldsymbol{0}, k(x, x'))\]

\[k(x,x') = \eta^2 \exp\left\{ -\frac{1}{2}\frac{(x - x')^2}{\ell_\mathrm{SE}^2}\right\}\left( 1 + \frac{\sqrt{3(x - x')^2}}{\ell_\mathrm{M}}\right) \exp\left\{ -\frac{\sqrt{3(x - x')^2}}{\ell_\mathrm{M}}\right\}\]

\[\ell_\mathrm{SE} \sim \mathrm{InvGamma}(5,5)\]

\[\ell_\mathrm{M} \sim \mathrm{InvGamma}(5,5)\]

\[\eta \sim \mathcal{N}^+(0,1)\]

\[\sigma_i \sim \mathcal{N}^+(\textrm{stderr}(y_i), \mathrm{Var}(\textrm{stderr}(\boldsymbol{y})))\]

9.1 MCMC Results

  variable      mean    median        sd       mad        q5        q95
 eta        0.002755  0.002114  0.002895  0.001097  0.000975   0.006462
 ell_SE    57.783790 51.838600 25.568304 18.683873 30.500890 104.864100
 ell_M     60.351649 57.814150 15.096787 12.721375 41.445750  88.143095
 sigma[1]   0.000086  0.000086  0.000009  0.000009  0.000071   0.000100
 f_star[1]  0.060711  0.060712  0.000092  0.000091  0.060559   0.060860
     rhat ess_bulk ess_tail
 1.000629     9063     4920
 1.000249     8588     4476
 1.000243     8519     5503
 1.001430    11979     4810
 0.999968     7345     7879

9.2 MCMC Plots

9.3 Posterior Predictive Samples

9.4 PSD

10 SE + Matern 3/2 + QP kernel

\[y_i \sim \mathcal{N}(f(x_i), \sigma_i^2)\]

\[f \sim \mathcal{GP}(\boldsymbol{0}, k(x, x'))\]

\[k(x,x') = \eta^2 \left[ \exp\left\{ -\frac{2 \sin^2\left( \pi\frac{\sqrt{(x - x')^2}}{T}\right)}{\ell_\mathrm{P}^2}\right\} + \exp\left\{ -\frac{1}{2}\frac{(x - x')^2}{\ell_\mathrm{SE}^2}\right\} + \left( 1 + \frac{\sqrt{3(x - x')^2}}{\ell_\mathrm{M}}\right) \exp\left\{ -\frac{\sqrt{3(x - x')^2}}{\ell_\mathrm{M}}\right\} \right]\]

\[\ell_\mathrm{P} \sim \mathrm{InvGamma}(5,5)\]

\[\ell_\mathrm{SE} \sim \mathrm{InvGamma}(5,5)\]

\[\ell_\mathrm{M} \sim \mathrm{InvGamma}(5,5)\]

\[\eta \sim \mathcal{N}^+(0,1)\]

\[T \sim \mathcal{U}[\textrm{minimum gap in x}, \textrm{range of x}]\]

\[\sigma_i \sim \mathcal{N}^+(\textrm{stderr}(y_i), \mathrm{Var}(\textrm{stderr}(\boldsymbol{y})))\]

10.1 MCMC Results

 variable     mean   median      sd     mad      q5      q95   rhat ess_bulk
   eta      0.0004   0.0003  0.0002  0.0001  0.0001   0.0007 1.0082     6068
   ell_SE  29.8219  26.0844 15.6114  9.3955 15.4976  53.6746 1.0136      298
   ell_M   28.1432  27.4783  6.7953  6.2933 18.6478  39.9603 1.0051     2527
   ell_P    2.3966   2.1037  1.2479  0.7938  1.1528   4.5984 1.0094     8752
   T      135.9911 146.0825 48.0893 47.9465 37.6883 193.9104 1.0524       51
 ess_tail
     5788
     4114
     5610
     4958
       11

10.2 MCMC Plots

10.3 Posterior Predictive Samples

10.4 PSD